Edymnion:There is a very easy way to determine if the heliocentric plane is parallel or perpendicular to the galactic plane, and fortunately you can walk outside right now and see it (which is a rare occurrence to have it line up so nicely).

You can see the band of the milky way behind the sun (which was part of the whole mayan doomsday thing, the galactic alignment). That shows that the solar system is parallel to the galactic plane as you can see the line formed by the Eath, the Sun, and the center of the galaxy. If it were perpendicular, then at no point could this alignment occur.

And if we want to get technical, the Eart/Sun/Galactic Center alignment is pretty darned routine, its just rare that we get all of the tilts and timing to line up for us to see it so easily.

I'm not sure about you, but I have trouble seeing the other stars during the daytime.